Abstract:Let be a bounded domain in , let H be a matrix
with . Let and consider the functional
over the class of Lipschitz functions
from satisfying affine boundary condition F.
It can be shown by convex integration that there exists
and with
. In this paper we begin the study of the asymptotics
of for such F.
This is the simplest minimisation problem involving surface energy in which
we can hope to see the effects of convex integration solutions. The only
known lower bounds are
. In this paper we
link the behavior of to the minimum of over a suitable
class of piecewise affine functions. Let be a
triangulation of by triangles of diameter less than h and
let denote the class of continuous functions that are piecewise affine
on a triangulation . For function
let be the interpolant, i.e.
the function we obtain by defining to be
the affine interpolation of u on the corners of . We show that
if for some small there exists
with
then for the interpolant
satisfies .

Note that it is conjectured that and it is trivial that
so we reduce the problem
of non-trivial lower bounds on to the problem of non-trivial lower bounds
on . This latter point will be addressed in a forthcoming paper.